LMFDB - Embedded newform 448.3.l.b.209.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [448,3,Mod(209,448)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(448, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 3, 2])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("448.209"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	448.l (of order $4$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [56] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$12.2071158433$
Analytic rank:	$0$
Dimension:	$56$
Relative dimension:	$28$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 112)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			209.1
Character	$\chi$	$=$	448.209
Dual form			448.3.l.b.433.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-3.98373 + 3.98373i) q^{3} +(2.85829 + 2.85829i) q^{5} +(6.99861 - 0.139610i) q^{7} -22.7401i q^{9} +(7.34382 - 7.34382i) q^{11} +(7.74138 - 7.74138i) q^{13} -22.7733 q^{15} -16.9118i q^{17} +(9.10220 - 9.10220i) q^{19} +(-27.3244 + 28.4367i) q^{21} -38.7551i q^{23} -8.66033i q^{25} +(54.7370 + 54.7370i) q^{27} +(14.6486 + 14.6486i) q^{29} +8.79153i q^{31} +58.5115i q^{33} +(20.4031 + 19.6050i) q^{35} +(-5.59597 + 5.59597i) q^{37} +61.6791i q^{39} -0.648870 q^{41} +(-11.6035 + 11.6035i) q^{43} +(64.9980 - 64.9980i) q^{45} +5.84809i q^{47} +(48.9610 - 1.95416i) q^{49} +(67.3721 + 67.3721i) q^{51} +(34.7291 - 34.7291i) q^{53} +41.9816 q^{55} +72.5213i q^{57} +(-4.12592 - 4.12592i) q^{59} +(-32.6840 + 32.6840i) q^{61} +(-3.17476 - 159.149i) q^{63} +44.2542 q^{65} +(-74.0032 - 74.0032i) q^{67} +(154.390 + 154.390i) q^{69} -24.7275i q^{71} -95.7422 q^{73} +(34.5004 + 34.5004i) q^{75} +(50.3712 - 52.4218i) q^{77} +43.7384 q^{79} -231.453 q^{81} +(-82.9808 + 82.9808i) q^{83} +(48.3389 - 48.3389i) q^{85} -116.712 q^{87} +93.1114 q^{89} +(53.0981 - 55.2597i) q^{91} +(-35.0230 - 35.0230i) q^{93} +52.0335 q^{95} +116.527i q^{97} +(-167.000 - 167.000i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q + 8 q^{15} - 20 q^{21} - 96 q^{29} + 100 q^{35} - 128 q^{37} + 72 q^{43} + 192 q^{49} + 128 q^{51} + 88 q^{53} - 444 q^{63} - 8 q^{65} - 440 q^{67} + 12 q^{77} + 8 q^{79} + 64 q^{81} + 96 q^{85} + 388 q^{91}+ \cdots - 8 q^{99}+O(q^{100}) $ 56 * q + 8 * q^15 - 20 * q^21 - 96 * q^29 + 100 * q^35 - 128 * q^37 + 72 * q^43 + 192 * q^49 + 128 * q^51 + 88 * q^53 - 444 * q^63 - 8 * q^65 - 440 * q^67 + 12 * q^77 + 8 * q^79 + 64 * q^81 + 96 * q^85 + 388 * q^91 + 32 * q^93 + 776 * q^95 - 8 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/448\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$197$
$\chi(n)$	$1$	$-1$	$e\left(\frac{3}{4}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−3.98373	+	3.98373i	−1.32791	+	1.32791i	−0.420716	+	0.907192i	$0.638221\pi$
$3$	−3.98373	+	3.98373i	−1.32791	+	1.32791i	−0.907192	+	0.420716i	$0.861779\pi$
$4$		0			0
$5$	2.85829	+	2.85829i	0.571658	+	0.571658i	0.932592	−	0.360933i	$-0.117542\pi$
$5$	2.85829	+	2.85829i	0.571658	+	0.571658i	−0.360933	+	0.932592i	$0.617542\pi$
$6$		0			0
$7$	6.99861	−	0.139610i	0.999801	−	0.0199444i
$7$	6.99861	−	0.139610i	0.999801	−	0.0199444i
$8$		0			0
$9$		−	22.7401i		−	2.52668i
$10$		0			0
$11$	7.34382	−	7.34382i	0.667620	−	0.667620i	−0.289544	−	0.957165i	$-0.593504\pi$
$11$	7.34382	−	7.34382i	0.667620	−	0.667620i	0.957165	+	0.289544i	$0.0935038\pi$
$12$		0			0
$13$	7.74138	−	7.74138i	0.595491	−	0.595491i	−0.343619	−	0.939109i	$-0.611653\pi$
$13$	7.74138	−	7.74138i	0.595491	−	0.595491i	0.939109	+	0.343619i	$0.111653\pi$
$14$		0			0
$15$	−22.7733			−1.51822
$16$		0			0
$17$		−	16.9118i		−	0.994813i	−0.867518	−	0.497407i	$-0.834286\pi$
$17$		−	16.9118i		−	0.994813i	0.867518	−	0.497407i	$-0.165714\pi$
$18$		0			0
$19$	9.10220	−	9.10220i	0.479063	−	0.479063i	−0.425769	−	0.904832i	$-0.639996\pi$
$19$	9.10220	−	9.10220i	0.479063	−	0.479063i	0.904832	+	0.425769i	$0.139996\pi$
$20$		0			0
$21$	−27.3244	+	28.4367i	−1.30116	+	1.35413i
$22$		0			0
$23$		−	38.7551i		−	1.68500i	−0.538694	−	0.842502i	$-0.681082\pi$
$23$		−	38.7551i		−	1.68500i	0.538694	−	0.842502i	$-0.318918\pi$
$24$		0			0
$25$		−	8.66033i		−	0.346413i
$26$		0			0
$27$	54.7370	+	54.7370i	2.02730	+	2.02730i
$28$		0			0
$29$	14.6486	+	14.6486i	0.505124	+	0.505124i	0.913026	−	0.407902i	$-0.133739\pi$
$29$	14.6486	+	14.6486i	0.505124	+	0.505124i	−0.407902	+	0.913026i	$0.633739\pi$
$30$		0			0
$31$			8.79153i			0.283598i	0.989895	+	0.141799i	$0.0452886\pi$
$31$			8.79153i			0.283598i	−0.989895	+	0.141799i	$0.954711\pi$
$32$		0			0
$33$			58.5115i			1.77308i
$34$		0			0
$35$	20.4031	+	19.6050i	0.582946	+	0.560143i
$36$		0			0
$37$	−5.59597	+	5.59597i	−0.151242	+	0.151242i	−0.778673	−	0.627430i	$-0.784107\pi$
$37$	−5.59597	+	5.59597i	−0.151242	+	0.151242i	0.627430	+	0.778673i	$0.284107\pi$
$38$		0			0
$39$			61.6791i			1.58151i
$40$		0			0
$41$	−0.648870			−0.0158261			−0.00791304	−	0.999969i	$-0.502519\pi$
$41$	−0.648870			−0.0158261			−0.00791304	+	0.999969i	$0.502519\pi$
$42$		0			0
$43$	−11.6035	+	11.6035i	−0.269849	+	0.269849i	−0.829039	−	0.559190i	$-0.811112\pi$
$43$	−11.6035	+	11.6035i	−0.269849	+	0.269849i	0.559190	+	0.829039i	$0.311112\pi$
$44$		0			0
$45$	64.9980	−	64.9980i	1.44440	−	1.44440i
$46$		0			0
$47$			5.84809i			0.124428i	0.998063	+	0.0622138i	$0.0198161\pi$
$47$			5.84809i			0.124428i	−0.998063	+	0.0622138i	$0.980184\pi$
$48$		0			0
$49$	48.9610	−	1.95416i	0.999204	−	0.0398808i
$50$		0			0
$51$	67.3721	+	67.3721i	1.32102	+	1.32102i
$52$		0			0
$53$	34.7291	−	34.7291i	0.655266	−	0.655266i	−0.298990	−	0.954256i	$-0.596650\pi$
$53$	34.7291	−	34.7291i	0.655266	−	0.655266i	0.954256	+	0.298990i	$0.0966497\pi$
$54$		0			0
$55$	41.9816			0.763301
$56$		0			0
$57$			72.5213i			1.27230i
$58$		0			0
$59$	−4.12592	−	4.12592i	−0.0699308	−	0.0699308i	0.671276	−	0.741207i	$-0.265746\pi$
$59$	−4.12592	−	4.12592i	−0.0699308	−	0.0699308i	−0.741207	+	0.671276i	$0.765746\pi$
$60$		0			0
$61$	−32.6840	+	32.6840i	−0.535804	+	0.535804i	−0.922294	−	0.386490i	$-0.873687\pi$
$61$	−32.6840	+	32.6840i	−0.535804	+	0.535804i	0.386490	+	0.922294i	$0.373687\pi$
$62$		0			0
$63$	−3.17476	−	159.149i	−0.0503931	−	2.52618i
$64$		0			0
$65$	44.2542			0.680835
$66$		0			0
$67$	−74.0032	−	74.0032i	−1.10453	−	1.10453i	−0.993857	−	0.110669i	$-0.964701\pi$
$67$	−74.0032	−	74.0032i	−1.10453	−	1.10453i	−0.110669	−	0.993857i	$-0.535299\pi$
$68$		0			0
$69$	154.390	+	154.390i	2.23753	+	2.23753i
$70$		0			0
$71$		−	24.7275i		−	0.348274i	−0.984721	−	0.174137i	$-0.944286\pi$
$71$		−	24.7275i		−	0.348274i	0.984721	−	0.174137i	$-0.0557136\pi$
$72$		0			0
$73$	−95.7422			−1.31154			−0.655769	−	0.754962i	$-0.727655\pi$
$73$	−95.7422			−1.31154			−0.655769	+	0.754962i	$0.727655\pi$
$74$		0			0
$75$	34.5004	+	34.5004i	0.460005	+	0.460005i
$76$		0			0
$77$	50.3712	−	52.4218i	0.654172	−	0.680803i
$78$		0			0
$79$	43.7384			0.553651			0.276826	−	0.960920i	$-0.410718\pi$
$79$	43.7384			0.553651			0.276826	+	0.960920i	$0.410718\pi$
$80$		0			0
$81$	−231.453			−2.85744
$82$		0			0
$83$	−82.9808	+	82.9808i	−0.999769	+	0.999769i	−1.00000		0.000231097i	$-0.999926\pi$
$83$	−82.9808	+	82.9808i	−0.999769	+	0.999769i	0.000231097		1.00000i	$0.499926\pi$
$84$		0			0
$85$	48.3389	−	48.3389i	0.568693	−	0.568693i
$86$		0			0
$87$	−116.712			−1.34152
$88$		0			0
$89$	93.1114			1.04620			0.523098	−	0.852273i	$-0.324776\pi$
$89$	93.1114			1.04620			0.523098	+	0.852273i	$0.324776\pi$
$90$		0			0
$91$	53.0981	−	55.2597i	0.583496	−	0.607249i
$92$		0			0
$93$	−35.0230	−	35.0230i	−0.376592	−	0.376592i
$94$		0			0
$95$	52.0335			0.547721
$96$		0			0
$97$			116.527i			1.20131i	0.799507	+	0.600657i	$0.205094\pi$
$97$			116.527i			1.20131i	−0.799507	+	0.600657i	$0.794906\pi$
$98$		0			0
$99$	−167.000	−	167.000i	−1.68686	−	1.68686i
$100$		0			0
$101$	−60.6954	−	60.6954i	−0.600945	−	0.600945i	0.339618	−	0.940563i	$-0.389702\pi$
$101$	−60.6954	−	60.6954i	−0.600945	−	0.600945i	−0.940563	+	0.339618i	$0.889702\pi$
$102$		0			0
$103$	−88.6335			−0.860520			−0.430260	−	0.902705i	$-0.641578\pi$
$103$	−88.6335			−0.860520			−0.430260	+	0.902705i	$0.641578\pi$
$104$		0			0
$105$	−159.381	+	3.17939i	−1.51792	+	0.0302799i
$106$		0			0
$107$	−99.3107	+	99.3107i	−0.928137	+	0.928137i	−0.997586	−	0.0694484i	$-0.977876\pi$
$107$	−99.3107	+	99.3107i	−0.928137	+	0.928137i	0.0694484	+	0.997586i	$0.477876\pi$
$108$		0			0
$109$	−44.7537	−	44.7537i	−0.410584	−	0.410584i	0.471358	−	0.881942i	$-0.343764\pi$
$109$	−44.7537	−	44.7537i	−0.410584	−	0.410584i	−0.881942	+	0.471358i	$0.843764\pi$
$110$		0			0
$111$		−	44.5856i		−	0.401672i
$112$		0			0
$113$	119.377			1.05643			0.528216	−	0.849110i	$-0.322861\pi$
$113$	119.377			1.05643			0.528216	+	0.849110i	$0.322861\pi$
$114$		0			0
$115$	110.773	−	110.773i	0.963246	−	0.963246i
$116$		0			0
$117$	−176.040	−	176.040i	−1.50462	−	1.50462i
$118$		0			0
$119$	−2.36107	−	118.359i	−0.0198409	−	0.994615i
$120$		0			0
$121$			13.1366i			0.108567i
$122$		0			0
$123$	2.58492	−	2.58492i	0.0210156	−	0.0210156i
$124$		0			0
$125$	96.2111	−	96.2111i	0.769688	−	0.769688i
$126$		0			0
$127$	71.4706			0.562761			0.281380	−	0.959596i	$-0.409208\pi$
$127$	71.4706			0.562761			0.281380	+	0.959596i	$0.409208\pi$
$128$		0			0
$129$		−	92.4503i		−	0.716669i
$130$		0			0
$131$	40.0674	−	40.0674i	0.305858	−	0.305858i	−0.537442	−	0.843301i	$-0.680609\pi$
$131$	40.0674	−	40.0674i	0.305858	−	0.305858i	0.843301	+	0.537442i	$0.180609\pi$
$132$		0			0
$133$	62.4320	−	64.9735i	0.469413	−	0.488522i
$134$		0			0
$135$			312.908i			2.31784i
$136$		0			0
$137$		−	42.7346i		−	0.311932i	−0.987762	−	0.155966i	$-0.950151\pi$
$137$		−	42.7346i		−	0.311932i	0.987762	−	0.155966i	$-0.0498489\pi$
$138$		0			0
$139$	123.602	+	123.602i	0.889226	+	0.889226i	0.994449	−	0.105223i	$-0.0335555\pi$
$139$	123.602	+	123.602i	0.889226	+	0.889226i	−0.105223	+	0.994449i	$0.533555\pi$
$140$		0			0
$141$	−23.2972	−	23.2972i	−0.165228	−	0.165228i
$142$		0			0
$143$		−	113.703i		−	0.795123i
$144$		0			0
$145$			83.7400i			0.577517i
$146$		0			0
$147$	−187.262	+	202.832i	−1.27389	+	1.37981i
$148$		0			0
$149$	45.5452	−	45.5452i	0.305673	−	0.305673i	−0.537556	−	0.843228i	$-0.680652\pi$
$149$	45.5452	−	45.5452i	0.305673	−	0.305673i	0.843228	+	0.537556i	$0.180652\pi$
$150$		0			0
$151$		−	79.4115i		−	0.525904i	−0.964809	−	0.262952i	$-0.915304\pi$
$151$		−	79.4115i		−	0.525904i	0.964809	−	0.262952i	$-0.0846961\pi$
$152$		0			0
$153$	−384.577			−2.51358
$154$		0			0
$155$	−25.1288	+	25.1288i	−0.162121	+	0.162121i
$156$		0			0
$157$	−146.600	+	146.600i	−0.933756	+	0.933756i	−0.997938	−	0.0641826i	$-0.979556\pi$
$157$	−146.600	+	146.600i	−0.933756	+	0.933756i	0.0641826	+	0.997938i	$0.479556\pi$
$158$		0			0
$159$			276.703i			1.74027i
$160$		0			0
$161$	−5.41061	−	271.232i	−0.0336063	−	1.68467i
$162$		0			0
$163$	83.6012	+	83.6012i	0.512891	+	0.512891i	0.915411	−	0.402520i	$-0.131866\pi$
$163$	83.6012	+	83.6012i	0.512891	+	0.512891i	−0.402520	+	0.915411i	$0.631866\pi$
$164$		0			0
$165$	−167.243	+	167.243i	−1.01359	+	1.01359i
$166$		0			0
$167$	36.5918			0.219113			0.109556	−	0.993981i	$-0.465057\pi$
$167$	36.5918			0.219113			0.109556	+	0.993981i	$0.465057\pi$
$168$		0			0
$169$			49.1421i			0.290782i
$170$		0			0
$171$	−206.985	−	206.985i	−1.21044	−	1.21044i
$172$		0			0
$173$	−15.0359	+	15.0359i	−0.0869129	+	0.0869129i	−0.749227	−	0.662314i	$-0.769575\pi$
$173$	−15.0359	+	15.0359i	−0.0869129	+	0.0869129i	0.662314	+	0.749227i	$0.269575\pi$
$174$		0			0
$175$	−1.20907	−	60.6103i	−0.00690899	−	0.346344i
$176$		0			0
$177$	32.8730			0.185723
$178$		0			0
$179$	195.329	+	195.329i	1.09123	+	1.09123i	0.995398	+	0.0958270i	$0.0305496\pi$
$179$	195.329	+	195.329i	1.09123	+	1.09123i	0.0958270	+	0.995398i	$0.469450\pi$
$180$		0			0
$181$	8.27670	+	8.27670i	0.0457276	+	0.0457276i	0.729601	−	0.683873i	$-0.239706\pi$
$181$	8.27670	+	8.27670i	0.0457276	+	0.0457276i	−0.683873	+	0.729601i	$0.739706\pi$
$182$		0			0
$183$		−	260.408i		−	1.42300i
$184$		0			0
$185$	−31.9898			−0.172918
$186$		0			0
$187$	−124.197	−	124.197i	−0.664157	−	0.664157i
$188$		0			0
$189$	390.724	+	375.441i	2.06732	+	1.98646i
$190$		0			0
$191$	255.184			1.33604			0.668021	−	0.744142i	$-0.267141\pi$
$191$	255.184			1.33604			0.668021	+	0.744142i	$0.267141\pi$
$192$		0			0
$193$	201.246			1.04273			0.521363	−	0.853335i	$-0.325424\pi$
$193$	201.246			1.04273			0.521363	+	0.853335i	$0.325424\pi$
$194$		0			0
$195$	−176.297	+	176.297i	−0.904086	+	0.904086i
$196$		0			0
$197$	−214.513	+	214.513i	−1.08890	+	1.08890i	−0.0932573	+	0.995642i	$0.529728\pi$
$197$	−214.513	+	214.513i	−1.08890	+	1.08890i	−0.995642	+	0.0932573i	$0.970272\pi$
$198$		0			0
$199$	24.0744			0.120977			0.0604885	−	0.998169i	$-0.480734\pi$
$199$	24.0744			0.120977			0.0604885	+	0.998169i	$0.480734\pi$
$200$		0			0
$201$	589.617			2.93342
$202$		0			0
$203$	104.565	+	100.475i	0.515098	+	0.494949i
$204$		0			0
$205$	−1.85466	−	1.85466i	−0.00904712	−	0.00904712i
$206$		0			0
$207$	−881.296			−4.25747
$208$		0			0
$209$		−	133.690i		−	0.639664i
$210$		0			0
$211$	183.834	+	183.834i	0.871251	+	0.871251i	0.992609	−	0.121358i	$-0.0387248\pi$
$211$	183.834	+	183.834i	0.871251	+	0.871251i	−0.121358	+	0.992609i	$0.538725\pi$
$212$		0			0
$213$	98.5075	+	98.5075i	0.462477	+	0.462477i
$214$		0			0
$215$	−66.3324			−0.308523
$216$		0			0
$217$	1.22739	+	61.5285i	0.00565617	+	0.283541i
$218$		0			0
$219$	381.411	−	381.411i	1.74160	−	1.74160i
$220$		0			0
$221$	−130.921	−	130.921i	−0.592402	−	0.592402i
$222$		0			0
$223$		−	194.715i		−	0.873161i	−0.899665	−	0.436580i	$-0.856189\pi$
$223$		−	194.715i		−	0.873161i	0.899665	−	0.436580i	$-0.143811\pi$
$224$		0			0
$225$	−196.937			−0.875277
$226$		0			0
$227$	151.403	−	151.403i	0.666973	−	0.666973i	−0.290042	−	0.957014i	$-0.593669\pi$
$227$	151.403	−	151.403i	0.666973	−	0.666973i	0.957014	+	0.290042i	$0.0936691\pi$
$228$		0			0
$229$	−225.367	−	225.367i	−0.984135	−	0.984135i	0.0157410	−	0.999876i	$-0.494989\pi$
$229$	−225.367	−	225.367i	−0.984135	−	0.984135i	−0.999876	+	0.0157410i	$0.994989\pi$
$230$		0			0
$231$	8.16882	+	409.499i	0.0353629	+	1.77272i
$232$		0			0
$233$		−	173.604i		−	0.745083i	−0.928016	−	0.372541i	$-0.878486\pi$
$233$		−	173.604i		−	0.745083i	0.928016	−	0.372541i	$-0.121514\pi$
$234$		0			0
$235$	−16.7156	+	16.7156i	−0.0711300	+	0.0711300i
$236$		0			0
$237$	−174.242	+	174.242i	−0.735198	+	0.735198i
$238$		0			0
$239$	−34.6861			−0.145130			−0.0725651	−	0.997364i	$-0.523119\pi$
$239$	−34.6861			−0.145130			−0.0725651	+	0.997364i	$0.523119\pi$
$240$		0			0
$241$			325.440i			1.35037i	0.737647	+	0.675186i	$0.235937\pi$
$241$			325.440i			1.35037i	−0.737647	+	0.675186i	$0.764063\pi$
$242$		0			0
$243$	429.412	−	429.412i	1.76713	−	1.76713i
$244$		0			0
$245$	145.530	+	134.359i	0.594002	+	0.548405i
$246$		0			0
$247$		−	140.927i		−	0.570555i
$248$		0			0
$249$		−	661.146i		−	2.65520i
$250$		0			0
$251$	−279.467	−	279.467i	−1.11341	−	1.11341i	−0.992686	−	0.120727i	$-0.961478\pi$
$251$	−279.467	−	279.467i	−1.11341	−	1.11341i	−0.120727	−	0.992686i	$-0.538522\pi$
$252$		0			0
$253$	−284.610	−	284.610i	−1.12494	−	1.12494i
$254$		0			0
$255$			385.138i			1.51035i
$256$		0			0
$257$		−	82.3722i		−	0.320514i	−0.987075	−	0.160257i	$-0.948768\pi$
$257$		−	82.3722i		−	0.320514i	0.987075	−	0.160257i	$-0.0512324\pi$
$258$		0			0
$259$	−38.3828	+	39.9453i	−0.148196	+	0.154229i
$260$		0			0
$261$	333.111	−	333.111i	1.27629	−	1.27629i
$262$		0			0
$263$		−	380.578i		−	1.44707i	−0.690290	−	0.723533i	$-0.742517\pi$
$263$		−	380.578i		−	1.44707i	0.690290	−	0.723533i	$-0.257483\pi$
$264$		0			0
$265$	198.532			0.749177
$266$		0			0
$267$	−370.930	+	370.930i	−1.38925	+	1.38925i
$268$		0			0
$269$	78.7424	−	78.7424i	0.292723	−	0.292723i	−0.545432	−	0.838155i	$-0.683634\pi$
$269$	78.7424	−	78.7424i	0.292723	−	0.292723i	0.838155	+	0.545432i	$0.183634\pi$
$270$		0			0
$271$			454.854i			1.67843i	0.543800	+	0.839215i	$0.316985\pi$
$271$			454.854i			1.67843i	−0.543800	+	0.839215i	$0.683015\pi$
$272$		0			0
$273$	8.61104	+	431.668i	0.0315423	+	1.58120i
$274$		0			0
$275$	−63.5999	−	63.5999i	−0.231273	−	0.231273i
$276$		0			0
$277$	−321.144	+	321.144i	−1.15937	+	1.15937i	−0.174753	+	0.984612i	$0.555913\pi$
$277$	−321.144	+	321.144i	−1.15937	+	1.15937i	−0.984612	+	0.174753i	$0.944087\pi$
$278$		0			0
$279$	199.921			0.716561
$280$		0			0
$281$			378.429i			1.34672i	0.739313	+	0.673362i	$0.235150\pi$
$281$			378.429i			1.34672i	−0.739313	+	0.673362i	$0.764850\pi$
$282$		0			0
$283$	310.879	+	310.879i	1.09851	+	1.09851i	0.994585	+	0.103927i	$0.0331408\pi$
$283$	310.879	+	310.879i	1.09851	+	1.09851i	0.103927	+	0.994585i	$0.466859\pi$
$284$		0			0
$285$	−207.287	+	207.287i	−0.727323	+	0.727323i
$286$		0			0
$287$	−4.54118	+	0.0905890i	−0.0158229	+	0.000315641i
$288$		0			0
$289$	2.99014			0.0103465
$290$		0			0
$291$	−464.213	−	464.213i	−1.59523	−	1.59523i
$292$		0			0
$293$	96.8340	+	96.8340i	0.330491	+	0.330491i	0.852773	−	0.522282i	$-0.174919\pi$
$293$	96.8340	+	96.8340i	0.330491	+	0.330491i	−0.522282	+	0.852773i	$0.674919\pi$
$294$		0			0
$295$		−	23.5861i		−	0.0799530i
$296$		0			0
$297$	803.957			2.70693
$298$		0			0
$299$	−300.018	−	300.018i	−1.00340	−	1.00340i
$300$		0			0
$301$	−79.5884	+	82.8283i	−0.264413	+	0.275177i
$302$		0			0
$303$	483.588			1.59600
$304$		0			0
$305$	−186.841			−0.612593
$306$		0			0
$307$	139.121	−	139.121i	0.453163	−	0.453163i	−0.443240	−	0.896403i	$-0.646171\pi$
$307$	139.121	−	139.121i	0.453163	−	0.453163i	0.896403	+	0.443240i	$0.146171\pi$
$308$		0			0
$309$	353.092	−	353.092i	1.14269	−	1.14269i
$310$		0			0
$311$	359.228			1.15507			0.577536	−	0.816365i	$-0.304014\pi$
$311$	359.228			1.15507			0.577536	+	0.816365i	$0.304014\pi$
$312$		0			0
$313$	−156.473			−0.499913			−0.249957	−	0.968257i	$-0.580416\pi$
$313$	−156.473			−0.499913			−0.249957	+	0.968257i	$0.580416\pi$
$314$		0			0
$315$	445.821	−	463.970i	1.41530	−	1.47292i
$316$		0			0
$317$	59.0354	+	59.0354i	0.186232	+	0.186232i	0.794065	−	0.607833i	$-0.207961\pi$
$317$	59.0354	+	59.0354i	0.186232	+	0.186232i	−0.607833	+	0.794065i	$0.707961\pi$
$318$		0			0
$319$	215.153			0.674462
$320$		0			0
$321$		−	791.253i		−	2.46496i
$322$		0			0
$323$	−153.935	−	153.935i	−0.476578	−	0.476578i
$324$		0			0
$325$	−67.0429	−	67.0429i	−0.206286	−	0.206286i
$326$		0			0
$327$	356.573			1.09044
$328$		0			0
$329$	0.816455	+	40.9285i	0.00248163	+	0.124403i
$330$		0			0
$331$	−382.660	+	382.660i	−1.15607	+	1.15607i	−0.170760	+	0.985313i	$0.554622\pi$
$331$	−382.660	+	382.660i	−1.15607	+	1.15607i	−0.985313	+	0.170760i	$0.945378\pi$
$332$		0			0
$333$	127.253	+	127.253i	0.382142	+	0.382142i
$334$		0			0
$335$		−	423.046i		−	1.26282i
$336$		0			0
$337$	140.848			0.417946			0.208973	−	0.977921i	$-0.432988\pi$
$337$	140.848			0.417946			0.208973	+	0.977921i	$0.432988\pi$
$338$		0			0
$339$	−475.565	+	475.565i	−1.40285	+	1.40285i
$340$		0			0
$341$	64.5634	+	64.5634i	0.189336	+	0.189336i
$342$		0			0
$343$	342.386	−	20.5119i	0.998210	−	0.0598013i
$344$		0			0
$345$			882.581i			2.55821i
$346$		0			0
$347$	−192.200	+	192.200i	−0.553891	+	0.553891i	−0.927561	−	0.373671i	$-0.878099\pi$
$347$	−192.200	+	192.200i	−0.553891	+	0.553891i	0.373671	+	0.927561i	$0.378099\pi$
$348$		0			0
$349$	100.387	−	100.387i	0.287642	−	0.287642i	−0.548505	−	0.836147i	$-0.684803\pi$
$349$	100.387	−	100.387i	0.287642	−	0.287642i	0.836147	+	0.548505i	$0.184803\pi$
$350$		0			0
$351$	847.479			2.41447
$352$		0			0
$353$		−	107.037i		−	0.303222i	−0.988440	−	0.151611i	$-0.951554\pi$
$353$		−	107.037i		−	0.303222i	0.988440	−	0.151611i	$-0.0484460\pi$
$354$		0			0
$355$	70.6784	−	70.6784i	0.199094	−	0.199094i
$356$		0			0
$357$	480.917	+	462.105i	1.34711	+	1.29441i
$358$		0			0
$359$			289.025i			0.805084i	0.915401	+	0.402542i	$0.131873\pi$
$359$			289.025i			0.805084i	−0.915401	+	0.402542i	$0.868127\pi$
$360$		0			0
$361$			195.300i			0.540997i
$362$		0			0
$363$	−52.3325	−	52.3325i	−0.144167	−	0.144167i
$364$		0			0
$365$	−273.659	−	273.659i	−0.749751	−	0.749751i
$366$		0			0
$367$			682.187i			1.85882i	0.369048	+	0.929410i	$0.379684\pi$
$367$			682.187i			1.85882i	−0.369048	+	0.929410i	$0.620316\pi$
$368$		0			0
$369$			14.7554i			0.0399875i
$370$		0			0
$371$	238.207	−	247.904i	0.642067	−	0.668205i
$372$		0			0
$373$	−224.714	+	224.714i	−0.602450	+	0.602450i	−0.940962	−	0.338512i	$-0.890076\pi$
$373$	−224.714	+	224.714i	−0.602450	+	0.602450i	0.338512	+	0.940962i	$0.390076\pi$
$374$		0			0
$375$			766.557i			2.04415i
$376$		0			0
$377$	226.801			0.601594
$378$		0			0
$379$	−86.4654	+	86.4654i	−0.228141	+	0.228141i	−0.811916	−	0.583775i	$-0.801575\pi$
$379$	−86.4654	+	86.4654i	−0.228141	+	0.228141i	0.583775	+	0.811916i	$0.301575\pi$
$380$		0			0
$381$	−284.719	+	284.719i	−0.747295	+	0.747295i
$382$		0			0
$383$		−	181.073i		−	0.472776i	−0.971659	−	0.236388i	$-0.924036\pi$
$383$		−	181.073i		−	0.472776i	0.971659	−	0.236388i	$-0.0759637\pi$
$384$		0			0
$385$	293.813	−	5.86107i	0.763149	−	0.0152236i
$386$		0			0
$387$	263.865	+	263.865i	0.681823	+	0.681823i
$388$		0			0
$389$	371.622	−	371.622i	0.955326	−	0.955326i	−0.0437177	−	0.999044i	$-0.513920\pi$
$389$	371.622	−	371.622i	0.955326	−	0.955326i	0.999044	+	0.0437177i	$0.0139202\pi$
$390$		0			0
$391$	−655.419			−1.67626
$392$		0			0
$393$			319.235i			0.812304i
$394$		0			0
$395$	125.017	+	125.017i	0.316499	+	0.316499i
$396$		0			0
$397$	70.5522	−	70.5522i	0.177713	−	0.177713i	−0.612645	−	0.790358i	$-0.709894\pi$
$397$	70.5522	−	70.5522i	0.177713	−	0.177713i	0.790358	+	0.612645i	$0.209894\pi$
$398$		0			0
$399$	10.1247	+	507.548i	0.0253753	+	1.27205i
$400$		0			0
$401$	273.378			0.681742			0.340871	−	0.940110i	$-0.389278\pi$
$401$	273.378			0.681742			0.340871	+	0.940110i	$0.389278\pi$
$402$		0			0
$403$	68.0586	+	68.0586i	0.168880	+	0.168880i
$404$		0			0
$405$	−661.560	−	661.560i	−1.63348	−	1.63348i
$406$		0			0
$407$			82.1916i			0.201945i
$408$		0			0
$409$	−93.7777			−0.229285			−0.114643	−	0.993407i	$-0.536572\pi$
$409$	−93.7777			−0.229285			−0.114643	+	0.993407i	$0.536572\pi$
$410$		0			0
$411$	170.243	+	170.243i	0.414217	+	0.414217i
$412$		0			0
$413$	−29.4517	−	28.2996i	−0.0713116	−	0.0685221i
$414$		0			0
$415$	−474.367			−1.14305
$416$		0			0
$417$	−984.796			−2.36162
$418$		0			0
$419$	208.427	−	208.427i	0.497438	−	0.497438i	−0.413201	−	0.910640i	$-0.635589\pi$
$419$	208.427	−	208.427i	0.497438	−	0.497438i	0.910640	+	0.413201i	$0.135589\pi$
$420$		0			0
$421$	90.0153	−	90.0153i	0.213813	−	0.213813i	−0.592072	−	0.805885i	$-0.701690\pi$
$421$	90.0153	−	90.0153i	0.213813	−	0.213813i	0.805885	+	0.592072i	$0.201690\pi$
$422$		0			0
$423$	132.986			0.314389
$424$		0			0
$425$	−146.462			−0.344617
$426$		0			0
$427$	−224.180	+	233.306i	−0.525011	+	0.546383i
$428$		0			0
$429$	452.960	+	452.960i	1.05585	+	1.05585i
$430$		0			0
$431$	−735.644			−1.70683			−0.853416	−	0.521231i	$-0.825473\pi$
$431$	−735.644			−1.70683			−0.853416	+	0.521231i	$0.825473\pi$
$432$		0			0
$433$			528.416i			1.22036i	0.792263	+	0.610180i	$0.208903\pi$
$433$			528.416i			1.22036i	−0.792263	+	0.610180i	$0.791097\pi$
$434$		0			0
$435$	−333.597	−	333.597i	−0.766890	−	0.766890i
$436$		0			0
$437$	−352.756	−	352.756i	−0.807223	−	0.807223i
$438$		0			0
$439$	−832.069			−1.89537			−0.947687	−	0.319202i	$-0.896585\pi$
$439$	−832.069			−1.89537			−0.947687	+	0.319202i	$0.896585\pi$
$440$		0			0
$441$	−44.4378	−	1113.38i	−0.100766	−	2.52467i
$442$		0			0
$443$	−211.932	+	211.932i	−0.478401	+	0.478401i	−0.904620	−	0.426219i	$-0.859845\pi$
$443$	−211.932	+	211.932i	−0.478401	+	0.478401i	0.426219	+	0.904620i	$0.359845\pi$
$444$		0			0
$445$	266.139	+	266.139i	0.598066	+	0.598066i
$446$		0			0
$447$			362.879i			0.811811i
$448$		0			0
$449$	86.2224			0.192032			0.0960161	−	0.995380i	$-0.469390\pi$
$449$	86.2224			0.192032			0.0960161	+	0.995380i	$0.469390\pi$
$450$		0			0
$451$	−4.76518	+	4.76518i	−0.0105658	+	0.0105658i
$452$		0			0
$453$	316.354	+	316.354i	0.698353	+	0.698353i
$454$		0			0
$455$	309.718	−	6.17836i	0.680699	−	0.0135788i
$456$		0			0
$457$		−	118.672i		−	0.259675i	−0.991535	−	0.129838i	$-0.958554\pi$
$457$		−	118.672i		−	0.259675i	0.991535	−	0.129838i	$-0.0414456\pi$
$458$		0			0
$459$	925.702	−	925.702i	2.01678	−	2.01678i
$460$		0			0
$461$	545.655	−	545.655i	1.18363	−	1.18363i	0.204837	−	0.978796i	$-0.434334\pi$
$461$	545.655	−	545.655i	1.18363	−	1.18363i	0.978796	−	0.204837i	$-0.0656663\pi$
$462$		0			0
$463$	112.656			0.243317			0.121659	−	0.992572i	$-0.461179\pi$
$463$	112.656			0.243317			0.121659	+	0.992572i	$0.461179\pi$
$464$		0			0
$465$		−	200.212i		−	0.430564i
$466$		0			0
$467$	−380.423	+	380.423i	−0.814609	+	0.814609i	−0.985321	−	0.170712i	$-0.945393\pi$
$467$	−380.423	+	380.423i	−0.814609	+	0.814609i	0.170712	+	0.985321i	$0.445393\pi$
$468$		0			0
$469$	−528.251	−	507.588i	−1.12634	−	1.08228i
$470$		0			0
$471$		−	1168.03i		−	2.47988i
$472$		0			0
$473$			170.428i			0.360313i
$474$		0			0
$475$	−78.8281	−	78.8281i	−0.165954	−	0.165954i
$476$		0			0
$477$	−789.745	−	789.745i	−1.65565	−	1.65565i
$478$		0			0
$479$		−	863.895i		−	1.80354i	−0.432217	−	0.901770i	$-0.642269\pi$
$479$		−	863.895i		−	1.80354i	0.432217	−	0.901770i	$-0.357731\pi$
$480$		0			0
$481$			86.6411i			0.180127i
$482$		0			0
$483$	1102.07	+	1058.96i	2.28171	+	2.19246i
$484$		0			0
$485$	−333.069	+	333.069i	−0.686741	+	0.686741i
$486$		0			0
$487$		−	345.108i		−	0.708640i	−0.935124	−	0.354320i	$-0.884712\pi$
$487$		−	345.108i		−	0.708640i	0.935124	−	0.354320i	$-0.115288\pi$
$488$		0			0
$489$	−666.088			−1.36214
$490$		0			0
$491$	129.810	−	129.810i	0.264379	−	0.264379i	−0.562452	−	0.826830i	$-0.690142\pi$
$491$	129.810	−	129.810i	0.264379	−	0.264379i	0.826830	+	0.562452i	$0.190142\pi$
$492$		0			0
$493$	247.735	−	247.735i	0.502504	−	0.502504i
$494$		0			0
$495$		−	954.667i		−	1.92862i
$496$		0			0
$497$	−3.45222	−	173.058i	−0.00694611	−	0.348205i
$498$		0			0
$499$	−223.327	−	223.327i	−0.447549	−	0.447549i	0.446990	−	0.894539i	$-0.352496\pi$
$499$	−223.327	−	223.327i	−0.447549	−	0.447549i	−0.894539	+	0.446990i	$0.852496\pi$
$500$		0			0
$501$	−145.772	+	145.772i	−0.290962	+	0.290962i
$502$		0			0
$503$	521.614			1.03701			0.518503	−	0.855076i	$-0.326489\pi$
$503$	521.614			1.03701			0.518503	+	0.855076i	$0.326489\pi$
$504$		0			0
$505$		−	346.970i		−	0.687070i
$506$		0			0
$507$	−195.769	−	195.769i	−0.386131	−	0.386131i
$508$		0			0
$509$	−438.781	+	438.781i	−0.862045	+	0.862045i	−0.991575	−	0.129530i	$-0.958653\pi$
$509$	−438.781	+	438.781i	−0.862045	+	0.862045i	0.129530	+	0.991575i	$0.458653\pi$
$510$		0			0
$511$	−670.062	+	13.3666i	−1.31128	+	0.0261578i
$512$		0			0
$513$	996.454			1.94240
$514$		0			0
$515$	−253.340	−	253.340i	−0.491923	−	0.491923i
$516$		0			0
$517$	42.9473	+	42.9473i	0.0830703	+	0.0830703i
$518$		0			0
$519$		−	119.798i		−	0.230825i
$520$		0			0
$521$	−281.992			−0.541252			−0.270626	−	0.962685i	$-0.587231\pi$
$521$	−281.992			−0.541252			−0.270626	+	0.962685i	$0.587231\pi$
$522$		0			0
$523$	79.8725	+	79.8725i	0.152720	+	0.152720i	0.779332	−	0.626612i	$-0.215559\pi$
$523$	79.8725	+	79.8725i	0.152720	+	0.152720i	−0.626612	+	0.779332i	$0.715559\pi$
$524$		0			0
$525$	246.271	+	236.638i	0.469088	+	0.450739i
$526$		0			0
$527$	148.681			0.282127
$528$		0			0
$529$	−972.956			−1.83924
$530$		0			0
$531$	−93.8239	+	93.8239i	−0.176693	+	0.176693i
$532$		0			0
$533$	−5.02315	+	5.02315i	−0.00942429	+	0.00942429i
$534$		0			0
$535$	−567.718			−1.06115
$536$		0			0
$537$	−1556.28			−2.89809
$538$		0			0
$539$	345.210	−	373.912i	0.640464	−	0.693714i
$540$		0			0
$541$	−44.8955	−	44.8955i	−0.0829861	−	0.0829861i	0.664395	−	0.747381i	$-0.268689\pi$
$541$	−44.8955	−	44.8955i	−0.0829861	−	0.0829861i	−0.747381	+	0.664395i	$0.768689\pi$
$542$		0			0
$543$	−65.9442			−0.121444
$544$		0			0
$545$		−	255.838i		−	0.469428i
$546$		0			0
$547$	−199.738	−	199.738i	−0.365152	−	0.365152i	0.500553	−	0.865706i	$-0.333130\pi$
$547$	−199.738	−	199.738i	−0.365152	−	0.365152i	−0.865706	+	0.500553i	$0.833130\pi$
$548$		0			0
$549$	743.239	+	743.239i	1.35381	+	1.35381i
$550$		0			0
$551$	266.669			0.483973
$552$		0			0
$553$	306.108	−	6.10635i	0.553541	−	0.0110422i
$554$		0			0
$555$	127.439	−	127.439i	0.229619	−	0.229619i
$556$		0			0
$557$	77.5628	+	77.5628i	0.139251	+	0.139251i	0.773296	−	0.634045i	$-0.218607\pi$
$557$	77.5628	+	77.5628i	0.139251	+	0.139251i	−0.634045	+	0.773296i	$0.718607\pi$
$558$		0			0
$559$			179.654i			0.321385i
$560$		0			0
$561$	989.537			1.76388
$562$		0			0
$563$	−197.910	+	197.910i	−0.351528	+	0.351528i	−0.860678	−	0.509150i	$-0.829960\pi$
$563$	−197.910	+	197.910i	−0.351528	+	0.351528i	0.509150	+	0.860678i	$0.329960\pi$
$564$		0			0
$565$	341.214	+	341.214i	0.603919	+	0.603919i
$566$		0			0
$567$	−1619.85	+	32.3132i	−2.85687	+	0.0569899i
$568$		0			0
$569$		−	569.566i		−	1.00100i	−0.865738	−	0.500498i	$-0.833150\pi$
$569$		−	569.566i		−	1.00100i	0.865738	−	0.500498i	$-0.166850\pi$
$570$		0			0
$571$	77.3963	−	77.3963i	0.135545	−	0.135545i	−0.636079	−	0.771624i	$-0.719445\pi$
$571$	77.3963	−	77.3963i	0.135545	−	0.135545i	0.771624	+	0.636079i	$0.219445\pi$
$572$		0			0
$573$	−1016.58	+	1016.58i	−1.77414	+	1.77414i
$574$		0			0
$575$	−335.632			−0.583708
$576$		0			0
$577$			897.407i			1.55530i	0.628698	+	0.777649i	$0.283588\pi$
$577$			897.407i			1.55530i	−0.628698	+	0.777649i	$0.716412\pi$
$578$		0			0
$579$	−801.709	+	801.709i	−1.38464	+	1.38464i
$580$		0			0
$581$	−569.165	+	592.335i	−0.979630	+	1.01951i
$582$		0			0
$583$		−	510.089i		−	0.874938i
$584$		0			0
$585$		−	1006.35i		−	1.72025i
$586$		0			0
$587$	−102.303	−	102.303i	−0.174281	−	0.174281i	0.614576	−	0.788857i	$-0.289327\pi$
$587$	−102.303	−	102.303i	−0.174281	−	0.174281i	−0.788857	+	0.614576i	$0.789327\pi$
$588$		0			0
$589$	80.0222	+	80.0222i	0.135861	+	0.135861i
$590$		0			0
$591$		−	1709.12i		−	2.89192i
$592$		0			0
$593$		−	687.227i		−	1.15890i	−0.815008	−	0.579449i	$-0.803268\pi$
$593$		−	687.227i		−	1.15890i	0.815008	−	0.579449i	$-0.196732\pi$
$594$		0			0
$595$	331.557	−	345.054i	0.557238	−	0.579922i
$596$		0			0
$597$	−95.9059	+	95.9059i	−0.160646	+	0.160646i
$598$		0			0
$599$			1083.21i			1.80837i	0.427145	+	0.904183i	$0.359519\pi$
$599$			1083.21i			1.80837i	−0.427145	+	0.904183i	$0.640481\pi$
$600$		0			0
$601$	1112.08			1.85038			0.925192	−	0.379500i	$-0.123904\pi$
$601$	1112.08			1.85038			0.925192	+	0.379500i	$0.123904\pi$
$602$		0			0
$603$	−1682.84	+	1682.84i	−2.79079	+	2.79079i
$604$		0			0
$605$	−37.5482	+	37.5482i	−0.0620631	+	0.0620631i
$606$		0			0
$607$		−	119.503i		−	0.196876i	−0.995143	−	0.0984378i	$-0.968615\pi$
$607$		−	119.503i		−	0.196876i	0.995143	−	0.0984378i	$-0.0313845\pi$
$608$		0			0
$609$	−816.822	+	16.2942i	−1.34125	+	0.0267557i
$610$		0			0
$611$	45.2723	+	45.2723i	0.0740954	+	0.0740954i
$612$		0			0
$613$	−176.246	+	176.246i	−0.287514	+	0.287514i	−0.836096	−	0.548582i	$-0.815168\pi$
$613$	−176.246	+	176.246i	−0.287514	+	0.287514i	0.548582	+	0.836096i	$0.315168\pi$
$614$		0			0
$615$	14.7769			0.0240275
$616$		0			0
$617$			850.125i			1.37784i	0.724839	+	0.688919i	$0.241914\pi$
$617$			850.125i			1.37784i	−0.724839	+	0.688919i	$0.758086\pi$
$618$		0			0
$619$	213.073	+	213.073i	0.344222	+	0.344222i	0.857952	−	0.513730i	$-0.171737\pi$
$619$	213.073	+	213.073i	0.344222	+	0.344222i	−0.513730	+	0.857952i	$0.671737\pi$
$620$		0			0
$621$	2121.34	−	2121.34i	3.41600	−	3.41600i
$622$		0			0
$623$	651.650	−	12.9993i	1.04599	−	0.0208657i
$624$		0			0
$625$	333.490			0.533584
$626$		0			0
$627$	532.584	+	532.584i	0.849416	+	0.849416i
$628$		0			0
$629$	94.6381	+	94.6381i	0.150458	+	0.150458i
$630$		0			0
$631$			326.063i			0.516741i	0.966046	+	0.258370i	$0.0831855\pi$
$631$			326.063i			0.516741i	−0.966046	+	0.258370i	$0.916814\pi$
$632$		0			0
$633$	−1464.69			−2.31388
$634$		0			0
$635$	204.284	+	204.284i	0.321707	+	0.321707i
$636$		0			0
$637$	363.898	−	394.154i	0.571268	−	0.618766i
$638$		0			0
$639$	−562.307			−0.879979
$640$		0			0
$641$	731.340			1.14094			0.570468	−	0.821320i	$-0.306762\pi$
$641$	731.340			1.14094			0.570468	+	0.821320i	$0.306762\pi$
$642$		0			0
$643$	257.820	−	257.820i	0.400964	−	0.400964i	−0.477609	−	0.878573i	$-0.658496\pi$
$643$	257.820	−	257.820i	0.400964	−	0.400964i	0.878573	+	0.477609i	$0.158496\pi$
$644$		0			0
$645$	264.250	−	264.250i	0.409690	−	0.409690i
$646$		0			0
$647$	−863.893			−1.33523			−0.667614	−	0.744507i	$-0.732684\pi$
$647$	−863.893			−1.33523			−0.667614	+	0.744507i	$0.732684\pi$
$648$		0			0
$649$	−60.6000			−0.0933744
$650$		0			0
$651$	−250.002	−	240.223i	−0.384028	−	0.369006i
$652$		0			0
$653$	−303.575	−	303.575i	−0.464892	−	0.464892i	0.435363	−	0.900255i	$-0.356620\pi$
$653$	−303.575	−	303.575i	−0.464892	−	0.464892i	−0.900255	+	0.435363i	$0.856620\pi$
$654$		0			0
$655$	229.049			0.349693
$656$		0			0
$657$			2177.19i			3.31384i
$658$		0			0
$659$	378.621	+	378.621i	0.574539	+	0.574539i	0.933394	−	0.358854i	$-0.116833\pi$
$659$	378.621	+	378.621i	0.574539	+	0.574539i	−0.358854	+	0.933394i	$0.616833\pi$
$660$		0			0
$661$	−407.564	−	407.564i	−0.616588	−	0.616588i	0.328067	−	0.944654i	$-0.393603\pi$
$661$	−407.564	−	407.564i	−0.616588	−	0.616588i	−0.944654	+	0.328067i	$0.893603\pi$
$662$		0			0
$663$	1043.11			1.57331
$664$		0			0
$665$	364.162	−	7.26442i	0.547612	−	0.0109239i
$666$		0			0
$667$	567.708	−	567.708i	0.851136	−	0.851136i
$668$		0			0
$669$	775.691	+	775.691i	1.15948	+	1.15948i
$670$		0			0
$671$			480.051i			0.715427i
$672$		0			0
$673$	−690.323			−1.02574			−0.512870	−	0.858466i	$-0.671418\pi$
$673$	−690.323			−1.02574			−0.512870	+	0.858466i	$0.671418\pi$
$674$		0			0
$675$	474.040	−	474.040i	0.702282	−	0.702282i
$676$		0			0
$677$	218.512	+	218.512i	0.322765	+	0.322765i	0.849827	−	0.527062i	$-0.176706\pi$
$677$	218.512	+	218.512i	0.322765	+	0.322765i	−0.527062	+	0.849827i	$0.676706\pi$
$678$		0			0
$679$	16.2684	+	815.529i	0.0239594	+	1.20107i
$680$		0			0
$681$			1206.29i			1.77136i
$682$		0			0
$683$	328.283	−	328.283i	0.480648	−	0.480648i	−0.424691	−	0.905339i	$-0.639617\pi$
$683$	328.283	−	328.283i	0.480648	−	0.480648i	0.905339	+	0.424691i	$0.139617\pi$
$684$		0			0
$685$	122.148	−	122.148i	0.178318	−	0.178318i
$686$		0			0
$687$	1795.60			2.61368
$688$		0			0
$689$		−	537.703i		−	0.780410i
$690$		0			0
$691$	−323.199	+	323.199i	−0.467727	+	0.467727i	−0.901177	−	0.433451i	$-0.857296\pi$
$691$	−323.199	+	323.199i	−0.467727	+	0.467727i	0.433451	+	0.901177i	$0.357296\pi$
$692$		0			0
$693$	−1192.08	−	1145.45i	−1.72017	−	1.65289i
$694$		0			0
$695$			706.584i			1.01667i
$696$		0			0
$697$			10.9736i			0.0157440i
$698$		0			0
$699$	691.592	+	691.592i	0.989402	+	0.989402i
$700$		0			0
$701$	704.778	+	704.778i	1.00539	+	1.00539i	0.999985	+	0.00540466i	$0.00172037\pi$
$701$	704.778	+	704.778i	1.00539	+	1.00539i	0.00540466	+	0.999985i	$0.498280\pi$
$702$		0			0
$703$			101.871i			0.144909i
$704$		0			0
$705$		−	133.180i		−	0.188908i
$706$		0			0
$707$	−433.257	−	416.310i	−0.612811	−	0.588840i
$708$		0			0
$709$	700.308	−	700.308i	0.987740	−	0.987740i	−0.0121858	−	0.999926i	$-0.503879\pi$
$709$	700.308	−	700.308i	0.987740	−	0.987740i	0.999926	+	0.0121858i	$0.00387896\pi$
$710$		0			0
$711$		−	994.619i		−	1.39890i
$712$		0			0
$713$	340.716			0.477863
$714$		0			0
$715$	324.995	−	324.995i	0.454539	−	0.454539i
$716$		0			0
$717$	138.180	−	138.180i	0.192720	−	0.192720i
$718$		0			0
$719$			25.6691i			0.0357012i	0.999841	+	0.0178506i	$0.00568232\pi$
$719$			25.6691i			0.0357012i	−0.999841	+	0.0178506i	$0.994318\pi$
$720$		0			0
$721$	−620.311	+	12.3742i	−0.860348	+	0.0171625i
$722$		0			0
$723$	−1296.46	−	1296.46i	−1.79317	−	1.79317i
$724$		0			0
$725$	126.862	−	126.862i	0.174982	−	0.174982i
$726$		0			0
$727$	−1039.69			−1.43010			−0.715052	−	0.699072i	$-0.753597\pi$
$727$	−1039.69			−1.43010			−0.715052	+	0.699072i	$0.753597\pi$
$728$		0			0
$729$			1338.24i			1.83573i
$730$		0			0
$731$	196.236	+	196.236i	0.268449	+	0.268449i
$732$		0			0
$733$	−580.702	+	580.702i	−0.792226	+	0.792226i	−0.981856	−	0.189630i	$-0.939271\pi$
$733$	−580.702	+	580.702i	−0.792226	+	0.792226i	0.189630	+	0.981856i	$0.439271\pi$
$734$		0			0
$735$	−1115.00	+	44.5026i	−1.51701	+	0.0605478i
$736$		0			0
$737$	−1086.93			−1.47481
$738$		0			0
$739$	194.170	+	194.170i	0.262747	+	0.262747i	0.826169	−	0.563422i	$-0.190516\pi$
$739$	194.170	+	194.170i	0.262747	+	0.262747i	−0.563422	+	0.826169i	$0.690516\pi$
$740$		0			0
$741$	561.415	+	561.415i	0.757645	+	0.757645i
$742$		0			0
$743$		−	827.119i		−	1.11322i	−0.830775	−	0.556608i	$-0.812103\pi$
$743$		−	827.119i		−	1.11322i	0.830775	−	0.556608i	$-0.187897\pi$
$744$		0			0
$745$	260.363			0.349481
$746$		0			0
$747$	1887.00	+	1887.00i	2.52610	+	2.52610i
$748$		0			0
$749$	−681.172	+	708.901i	−0.909441	+	0.946464i
$750$		0			0
$751$	920.918			1.22626			0.613128	−	0.789984i	$-0.289911\pi$
$751$	920.918			1.22626			0.613128	+	0.789984i	$0.289911\pi$
$752$		0			0
$753$	2226.64			2.95702
$754$		0			0
$755$	226.981	−	226.981i	0.300638	−	0.300638i
$756$		0			0
$757$	415.892	−	415.892i	0.549396	−	0.549396i	−0.376870	−	0.926266i	$-0.623000\pi$
$757$	415.892	−	415.892i	0.549396	−	0.549396i	0.926266	+	0.376870i	$0.123000\pi$
$758$		0			0
$759$	2267.62			2.98764
$760$		0			0
$761$	−1186.17			−1.55871			−0.779353	−	0.626586i	$-0.784452\pi$
$761$	−1186.17			−1.55871			−0.779353	+	0.626586i	$0.784452\pi$
$762$		0			0
$763$	−319.461	−	306.965i	−0.418691	−	0.402314i
$764$		0			0
$765$	−1099.23	−	1099.23i	−1.43691	−	1.43691i
$766$		0			0
$767$	−63.8806			−0.0832863
$768$		0			0
$769$		−	194.962i		−	0.253527i	−0.991933	−	0.126764i	$-0.959541\pi$
$769$		−	194.962i		−	0.253527i	0.991933	−	0.126764i	$-0.0404589\pi$
$770$		0			0
$771$	328.148	+	328.148i	0.425614	+	0.425614i
$772$		0			0
$773$	264.574	+	264.574i	0.342269	+	0.342269i	0.857220	−	0.514951i	$-0.172190\pi$
$773$	264.574	+	264.574i	0.342269	+	0.342269i	−0.514951	+	0.857220i	$0.672190\pi$
$774$		0			0
$775$	76.1376			0.0982420
$776$		0			0
$777$	−6.22462	−	312.037i	−0.00801110	−	0.401592i
$778$		0			0
$779$	−5.90614	+	5.90614i	−0.00758170	+	0.00758170i
$780$		0			0
$781$	−181.594	−	181.594i	−0.232515	−	0.232515i
$782$		0			0
$783$			1603.64i			2.04807i
$784$		0			0
$785$	−838.049			−1.06758
$786$		0			0
$787$	239.581	−	239.581i	0.304424	−	0.304424i	−0.538318	−	0.842742i	$-0.680940\pi$
$787$	239.581	−	239.581i	0.304424	−	0.304424i	0.842742	+	0.538318i	$0.180940\pi$
$788$		0			0
$789$	1516.12	+	1516.12i	1.92157	+	1.92157i
$790$		0			0
$791$	835.472	−	16.6663i	1.05622	−	0.0210699i
$792$		0			0
$793$			506.039i			0.638132i
$794$		0			0
$795$	−790.897	+	790.897i	−0.994839	+	0.994839i
$796$		0			0
$797$	554.730	−	554.730i	0.696022	−	0.696022i	−0.267528	−	0.963550i	$-0.586207\pi$
$797$	554.730	−	554.730i	0.696022	−	0.696022i	0.963550	+	0.267528i	$0.0862067\pi$
$798$		0			0
$799$	98.9019			0.123782
$800$		0			0
$801$		−	2117.37i		−	2.64340i
$802$		0			0
$803$	−703.114	+	703.114i	−0.875609	+	0.875609i
$804$		0			0
$805$	759.794	−	790.724i	0.943843	−	0.982266i
$806$		0			0
$807$			627.376i			0.777418i
$808$		0			0
$809$		−	876.722i		−	1.08371i	−0.840472	−	0.541855i	$-0.817722\pi$
$809$		−	876.722i		−	1.08371i	0.840472	−	0.541855i	$-0.182278\pi$
$810$		0			0
$811$	798.902	+	798.902i	0.985083	+	0.985083i	0.999890	−	0.0148077i	$-0.00471362\pi$
$811$	798.902	+	798.902i	0.985083	+	0.985083i	−0.0148077	+	0.999890i	$0.504714\pi$
$812$		0			0
$813$	−1812.02	−	1812.02i	−2.22880	−	2.22880i
$814$		0			0
$815$			477.913i			0.586397i
$816$		0			0
$817$			211.235i			0.258549i
$818$		0			0
$819$	−1256.61	−	1207.46i	−1.53433	−	1.47431i
$820$		0			0
$821$	488.321	−	488.321i	0.594788	−	0.594788i	−0.344133	−	0.938921i	$-0.611827\pi$
$821$	488.321	−	488.321i	0.594788	−	0.594788i	0.938921	+	0.344133i	$0.111827\pi$
$822$		0			0
$823$			481.599i			0.585175i	0.956239	+	0.292588i	$0.0945163\pi$
$823$			481.599i			0.585175i	−0.956239	+	0.292588i	$0.905484\pi$
$824$		0			0
$825$	506.729			0.614218
$826$		0			0
$827$	−269.789	+	269.789i	−0.326226	+	0.326226i	−0.851149	−	0.524924i	$-0.824094\pi$
$827$	−269.789	+	269.789i	−0.326226	+	0.326226i	0.524924	+	0.851149i	$0.324094\pi$
$828$		0			0
$829$	1021.06	−	1021.06i	1.23168	−	1.23168i	0.268357	−	0.963319i	$-0.413519\pi$
$829$	1021.06	−	1021.06i	1.23168	−	1.23168i	0.963319	−	0.268357i	$-0.0864808\pi$
$830$		0			0
$831$		−	2558.70i		−	3.07906i
$832$		0			0
$833$	−33.0484	−	828.020i	−0.0396739	−	0.994022i
$834$		0			0
$835$	104.590	+	104.590i	0.125258	+	0.125258i
$836$		0			0
$837$	−481.222	+	481.222i	−0.574936	+	0.574936i
$838$		0			0
$839$	770.486			0.918338			0.459169	−	0.888349i	$-0.348147\pi$
$839$	770.486			0.918338			0.459169	+	0.888349i	$0.348147\pi$
$840$		0			0
$841$		−	411.837i		−	0.489699i
$842$		0			0
$843$	−1507.56	−	1507.56i	−1.78833	−	1.78833i
$844$		0			0
$845$	−140.462	+	140.462i	−0.166228	+	0.166228i
$846$		0			0
$847$	1.83400	+	91.9378i	0.00216529	+	0.108545i
$848$		0			0
$849$	−2476.91			−2.91745
$850$		0			0
$851$	216.872	+	216.872i	0.254844	+	0.254844i
$852$		0			0
$853$	−593.299	−	593.299i	−0.695544	−	0.695544i	0.267902	−	0.963446i	$-0.413670\pi$
$853$	−593.299	−	593.299i	−0.695544	−	0.695544i	−0.963446	+	0.267902i	$0.913670\pi$
$854$		0			0
$855$		−	1183.25i		−	1.38392i
$856$		0			0
$857$	−525.392			−0.613060			−0.306530	−	0.951861i	$-0.599168\pi$
$857$	−525.392			−0.613060			−0.306530	+	0.951861i	$0.599168\pi$
$858$		0			0
$859$	−738.470	−	738.470i	−0.859686	−	0.859686i	0.131615	−	0.991301i	$-0.457984\pi$
$859$	−738.470	−	738.470i	−0.859686	−	0.859686i	−0.991301	+	0.131615i	$0.957984\pi$
$860$		0			0
$861$	17.7300	−	18.4517i	0.0205923	−	0.0214306i
$862$		0			0
$863$	−381.348			−0.441887			−0.220943	−	0.975287i	$-0.570914\pi$
$863$	−381.348			−0.441887			−0.220943	+	0.975287i	$0.570914\pi$
$864$		0			0
$865$	−85.9541			−0.0993690
$866$		0			0
$867$	−11.9119	+	11.9119i	−0.0137392	+	0.0137392i
$868$		0			0
$869$	321.207	−	321.207i	0.369629	−	0.369629i
$870$		0			0
$871$	−1145.77			−1.31547
$872$		0			0
$873$	2649.85			3.03534
$874$		0			0
$875$	659.911	−	686.776i	0.754184	−	0.784886i
$876$		0			0
$877$	847.709	+	847.709i	0.966601	+	0.966601i	0.999460	−	0.0328586i	$-0.0104611\pi$
$877$	847.709	+	847.709i	0.966601	+	0.966601i	−0.0328586	+	0.999460i	$0.510461\pi$
$878$		0			0
$879$	−771.520			−0.877725
$880$		0			0
$881$			873.115i			0.991050i	0.868594	+	0.495525i	$0.165024\pi$
$881$			873.115i			0.991050i	−0.868594	+	0.495525i	$0.834976\pi$
$882$		0			0
$883$	−439.716	−	439.716i	−0.497980	−	0.497980i	0.412829	−	0.910809i	$-0.364541\pi$
$883$	−439.716	−	439.716i	−0.497980	−	0.497980i	−0.910809	+	0.412829i	$0.864541\pi$
$884$		0			0
$885$	93.9607	+	93.9607i	0.106170	+	0.106170i
$886$		0			0
$887$	830.146			0.935903			0.467952	−	0.883754i	$-0.344992\pi$
$887$	830.146			0.935903			0.467952	+	0.883754i	$0.344992\pi$
$888$		0			0
$889$	500.195	−	9.97805i	0.562649	−	0.0112239i
$890$		0			0
$891$	−1699.75	+	1699.75i	−1.90769	+	1.90769i
$892$		0			0
$893$	53.2305	+	53.2305i	0.0596086	+	0.0596086i
$894$		0			0
$895$			1116.62i			1.24762i
$896$		0			0
$897$	2390.38			2.66486
$898$		0			0
$899$	−128.784	+	128.784i	−0.143252	+	0.143252i
$900$		0			0
$901$	−587.333	−	587.333i	−0.651868	−	0.651868i
$902$		0			0
$903$	−12.9070	−	647.024i	−0.0142935	−	0.716527i
$904$		0			0
$905$			47.3145i			0.0522812i
$906$		0			0
$907$	997.246	−	997.246i	1.09950	−	1.09950i	0.105030	−	0.994469i	$-0.466506\pi$
$907$	997.246	−	997.246i	1.09950	−	1.09950i	0.994469	−	0.105030i	$-0.0334940\pi$
$908$		0			0
$909$	−1380.22	+	1380.22i	−1.51840	+	1.51840i
$910$		0			0
$911$	362.386			0.397789			0.198894	−	0.980021i	$-0.436265\pi$
$911$	362.386			0.397789			0.198894	+	0.980021i	$0.436265\pi$
$912$		0			0
$913$			1218.79i			1.33493i
$914$		0			0
$915$	744.323	−	744.323i	0.813468	−	0.813468i
$916$		0			0
$917$	274.822	−	286.010i	0.299697	−	0.311898i
$918$		0			0
$919$			916.223i			0.996978i	0.866896	+	0.498489i	$0.166112\pi$
$919$			916.223i			0.996978i	−0.866896	+	0.498489i	$0.833888\pi$
$920$		0			0
$921$			1108.44i			1.20352i
$922$		0			0
$923$	−191.425	−	191.425i	−0.207394	−	0.207394i
$924$		0			0
$925$	48.4630	+	48.4630i	0.0523924	+	0.0523924i
$926$		0			0
$927$			2015.54i			2.17426i
$928$		0			0
$929$			1362.91i			1.46707i	0.679652	+	0.733535i	$0.262131\pi$
$929$			1362.91i			1.46707i	−0.679652	+	0.733535i	$0.737869\pi$
$930$		0			0
$931$	427.866	−	463.440i	0.459577	−	0.497787i
$932$		0			0
$933$	−1431.06	+	1431.06i	−1.53383	+	1.53383i
$934$		0			0
$935$		−	709.985i		−	0.759342i
$936$		0			0
$937$	666.821			0.711656			0.355828	−	0.934552i	$-0.384199\pi$
$937$	666.821			0.711656			0.355828	+	0.934552i	$0.384199\pi$
$938$		0			0
$939$	623.345	−	623.345i	0.663839	−	0.663839i
$940$		0			0
$941$	256.017	−	256.017i	0.272069	−	0.272069i	−0.557863	−	0.829933i	$-0.688379\pi$
$941$	256.017	−	256.017i	0.272069	−	0.272069i	0.829933	+	0.557863i	$0.188379\pi$
$942$		0			0
$943$			25.1470i			0.0266670i
$944$		0			0
$945$	43.6853	+	2189.92i	0.0462278	+	2.31738i
$946$		0			0
$947$	466.545	+	466.545i	0.492655	+	0.492655i	0.909142	−	0.416487i	$-0.136739\pi$
$947$	466.545	+	466.545i	0.492655	+	0.492655i	−0.416487	+	0.909142i	$0.636739\pi$
$948$		0			0
$949$	−741.177	+	741.177i	−0.781008	+	0.781008i
$950$		0			0
$951$	−470.362			−0.494597
$952$		0			0
$953$		−	731.374i		−	0.767444i	−0.923449	−	0.383722i	$-0.874642\pi$
$953$		−	731.374i		−	0.767444i	0.923449	−	0.383722i	$-0.125358\pi$
$954$		0			0
$955$	729.391	+	729.391i	0.763760	+	0.763760i
$956$		0			0
$957$	−857.112	+	857.112i	−0.895624	+	0.895624i
$958$		0			0
$959$	−5.96620	−	299.083i	−0.00622127	−	0.311869i
$960$		0			0
$961$	883.709			0.919572
$962$		0			0
$963$	2258.34	+	2258.34i	2.34511	+	2.34511i
$964$		0			0
$965$	575.220	+	575.220i	0.596083	+	0.596083i
$966$		0			0
$967$			193.963i			0.200582i	0.994958	+	0.100291i	$0.0319774\pi$
$967$			193.963i			0.200582i	−0.994958	+	0.100291i	$0.968023\pi$
$968$		0			0
$969$	1226.47			1.26570
$970$		0			0
$971$	−974.055	−	974.055i	−1.00315	−	1.00315i	−0.999995	−	0.00315102i	$-0.998997\pi$
$971$	−974.055	−	974.055i	−1.00315	−	1.00315i	−0.00315102	−	0.999995i	$-0.501003\pi$
$972$		0			0
$973$	882.301	+	847.789i	0.906784	+	0.871314i
$974$		0			0
$975$	534.161			0.547858
$976$		0			0
$977$	469.025			0.480067			0.240033	−	0.970765i	$-0.422842\pi$
$977$	469.025			0.480067			0.240033	+	0.970765i	$0.422842\pi$
$978$		0			0
$979$	683.793	−	683.793i	0.698461	−	0.698461i
$980$		0			0
$981$	−1017.71	+	1017.71i	−1.03742	+	1.03742i
$982$		0			0
$983$	−1695.60			−1.72493			−0.862463	−	0.506120i	$-0.831079\pi$
$983$	−1695.60			−1.72493			−0.862463	+	0.506120i	$0.831079\pi$
$984$		0			0
$985$	−1226.28			−1.24496
$986$		0			0
$987$	−166.300	−	159.795i	−0.168491	−	0.161900i
$988$		0			0
$989$	449.695	+	449.695i	0.454696	+	0.454696i
$990$		0			0
$991$	276.573			0.279085			0.139542	−	0.990216i	$-0.455437\pi$
$991$	276.573			0.279085			0.139542	+	0.990216i	$0.455437\pi$
$992$		0			0
$993$		−	3048.83i		−	3.07032i
$994$		0			0
$995$	68.8117	+	68.8117i	0.0691575	+	0.0691575i
$996$		0			0
$997$	−616.598	−	616.598i	−0.618453	−	0.618453i	0.326682	−	0.945134i	$-0.394070\pi$
$997$	−616.598	−	616.598i	−0.618453	−	0.618453i	−0.945134	+	0.326682i	$0.894070\pi$
$998$		0			0
$999$	−612.613			−0.613226

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.3.l.b.209.1		56
4.3	odd	2		112.3.l.b.13.22	yes	56
7.6	odd	2	inner	448.3.l.b.209.28		56
16.5	even	4	inner	448.3.l.b.433.28		56
16.11	odd	4		112.3.l.b.69.21	yes	56
28.27	even	2		112.3.l.b.13.21	✓	56
112.27	even	4		112.3.l.b.69.22	yes	56
112.69	odd	4	inner	448.3.l.b.433.1		56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.3.l.b.13.21	✓	56	28.27	even	2
112.3.l.b.13.22	yes	56	4.3	odd	2
112.3.l.b.69.21	yes	56	16.11	odd	4
112.3.l.b.69.22	yes	56	112.27	even	4
448.3.l.b.209.1		56	1.1	even	1	trivial
448.3.l.b.209.28		56	7.6	odd	2	inner
448.3.l.b.433.1		56	112.69	odd	4	inner
448.3.l.b.433.28		56	16.5	even	4	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	448.l (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-3.98373 + 3.98373i) q^{3} +(2.85829 + 2.85829i) q^{5} +(6.99861 - 0.139610i) q^{7} -22.7401i q^{9} +(7.34382 - 7.34382i) q^{11} +(7.74138 - 7.74138i) q^{13} -22.7733 q^{15} -16.9118i q^{17} +(9.10220 - 9.10220i) q^{19} +(-27.3244 + 28.4367i) q^{21} -38.7551i q^{23} -8.66033i q^{25} +(54.7370 + 54.7370i) q^{27} +(14.6486 + 14.6486i) q^{29} +8.79153i q^{31} +58.5115i q^{33} +(20.4031 + 19.6050i) q^{35} +(-5.59597 + 5.59597i) q^{37} +61.6791i q^{39} -0.648870 q^{41} +(-11.6035 + 11.6035i) q^{43} +(64.9980 - 64.9980i) q^{45} +5.84809i q^{47} +(48.9610 - 1.95416i) q^{49} +(67.3721 + 67.3721i) q^{51} +(34.7291 - 34.7291i) q^{53} +41.9816 q^{55} +72.5213i q^{57} +(-4.12592 - 4.12592i) q^{59} +(-32.6840 + 32.6840i) q^{61} +(-3.17476 - 159.149i) q^{63} +44.2542 q^{65} +(-74.0032 - 74.0032i) q^{67} +(154.390 + 154.390i) q^{69} -24.7275i q^{71} -95.7422 q^{73} +(34.5004 + 34.5004i) q^{75} +(50.3712 - 52.4218i) q^{77} +43.7384 q^{79} -231.453 q^{81} +(-82.9808 + 82.9808i) q^{83} +(48.3389 - 48.3389i) q^{85} -116.712 q^{87} +93.1114 q^{89} +(53.0981 - 55.2597i) q^{91} +(-35.0230 - 35.0230i) q^{93} +52.0335 q^{95} +116.527i q^{97} +(-167.000 - 167.000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q + 8 q^{15} - 20 q^{21} - 96 q^{29} + 100 q^{35} - 128 q^{37} + 72 q^{43} + 192 q^{49} + 128 q^{51} + 88 q^{53} - 444 q^{63} - 8 q^{65} - 440 q^{67} + 12 q^{77} + 8 q^{79} + 64 q^{81} + 96 q^{85} + 388 q^{91}+ \cdots - 8 q^{99}+O(q^{100}) \) 56 * q + 8 * q^15 - 20 * q^21 - 96 * q^29 + 100 * q^35 - 128 * q^37 + 72 * q^43 + 192 * q^49 + 128 * q^51 + 88 * q^53 - 444 * q^63 - 8 * q^65 - 440 * q^67 + 12 * q^77 + 8 * q^79 + 64 * q^81 + 96 * q^85 + 388 * q^91 + 32 * q^93 + 776 * q^95 - 8 * q^99

Introduction

L-functions

Modular forms

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